Maximal closed connected subgroups of positive dimension in $\mathrm{SL}(n,\mathbb{R})$ are parabolic, the normalizer of a connected semisimple subgroup, or the normalizer of a maximal torus.   There are finitely many conjugation classes of each type, and so you could try to work out the maximal ones.

**General References**

According to the MathSciNet review (see also MO answers [here][1] and [here][2]) this paper addresses the problem (without proof):

Komrakov, B. P. *Maximal subalgebras of real Lie algebras and a problem of Sophus Lie.* Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 528--532; translation in Soviet Math. Dokl. 41 (1990), no. 2, 269–273 (1991) 

Another paper that might be helpful is:

Mostow, G. D. *On maximal subgroups of real Lie groups.* 
Ann. of Math. (2) 74 1961 503–517. 


  [1]: http://mathoverflow.net/a/111532/12218
  [2]: http://mathoverflow.net/a/181867/12218
  [3]: http://www.maths.usyd.edu.au/u/murray/research/parabcc-p.pdf